Math books for advanced high school students I'm working in a program for teaching a group of students selected in a Olympiad competition. The program is aimed to acquaint the students with the diverse aspects of higher mathematics in a way which is attractive and giving them a  good viewpoint of the subject. So I'm looking for books which can be used in the courses of this program. Note that I don't want a university textbook that contains the elementary parts of an abstract subject and not giving a general insight. As an example I can mention "Arnold's lectures in Abel's theorem".
Please help me with your advices for book titles, Thanks!
 A: Complete set of Quantum magazine. It exists in good libraries and sometimes can be found on e-bay. The journal existed for 11 years (1990-2001) but apparently there are no enough "bright high school students" in the English speaking world to support this journal:-(
A: When I was in high school (some 20 years ago), there were translations of a series published by the MAA called the New Mathematical Library, which I enjoyed a lot.  They were mostly written by top mathematicians, and were extremely readable and inspiring.  
Aside from that, I second Proofs from THE BOOK which someone already mentioned.
A: The Pleasures of Counting by Tom (T W) Körner, Cambridge University Press, 1996,
ISBN13: 9780521568234.
Forget the IMO. This book is about the relevance of mathematics to the real world. Each chapter contains a piece of history (unfortunately often from wartime) in which mathematics provided a solution to a real problem.  This is then followed by a development of the relevant mathematical ideas.
A: I think one of good book is:
"The Art and Craft of Problem Solving" by Paul Zeitz
Also, there is a package with name: "Mathematical Olympiad Resources" which is contains many interesting books and best problems collected around the world. This source contains three text list, which introduces many interesting webpages and books. 
Also you need some good book about "probabilistic method" which nowadays is very powerful technique to attack some problems. There is some sources about it and some webpages with special problems. But I think you have to prepare the book "probabilistic method" by Noga Alon and prepare it with your own language for your students.
Finally, I suggest the good book "A First Step to Mathematical Olympiad Problems" by Derek Holton. This book has very nice approach to introduce, solve and generalize the problems.
A: Polya's Induction and Analogy in Mathematics (Volume I of Mathematics and Plausible Reasoning.
https://archive.org/details/Induction_And_Analogy_In_Mathematics_1_
http://www.amazon.com/Mathematics-Plausible-Reasoning-Volume-Induction/dp/0691025096 , used from $6.04.
A: 1- Shurman - Geometry of the Quintic
2- Weeks - The Shape of Space
3- Hatcher - Topology of Numbers
4- Arnold - Real Algebraic Geometry 
5- Stepanov - Arithmetic of Algebraic Curves
A: The Princeton Companion to Mathematics by Timothy Gowers discusses some very beautiful mathematics in a serious yet not too technical way.
A: As a high school student myself I discovered what mathematics was really about by reading Efimov's "higher geometry" which I bought pretty much at random. It deals mainly with hyperbolic and projective geometry, and in a completely axiomatic way. To me it was like a door to another world I never dreamed about. I highly recommend it.
A: A good book  to use for teaching students about  mathematical proofs is the  classic Hadamard's  monograph "Lectures on elementary geometry".  It should be translated in many languages by now.
A: M. Kac. Statistical independence in probability, analysis, and number theory. It is short (< 100 pages), requires little beyond a good background in one variable calculus, and introduces some real ideas and some interactions between apparently different areas. 
A: Etingof et al. 
Introduction to Representation Theory (Student Mathematical Library)
Which has a version online in Pavel's webpage
This book actually came to be from a class that Etingof gave a strong group of undergraduates I think in the SPURS program, most of them where from the Olympiads. 
A: The Cauchy-Shwarz Master Class, is a pretty fun book and not bogged down with technical detail.  Rather it is a ton of clever applications of this powerful inequality and its variants http://www.amazon.com/Cauchy-Schwarz-Master-Class-Introduction-Mathematical/dp/052154677X/ref=sr_1_1?ie=UTF8&qid=1415623270&sr=8-1&keywords=cauchy+schwarz 
A: I'd also suggest "Mathematical Omnibus: Thirty Lectures on Classic Mathematics". 
As it is said on its back cover: 

The book consists of thirty lectures on diverse topics, covering much of the mathematical landscape rather than focusing on one area. The reader will learn numerous results that often belong to neither the standard undergraduate nor graduate curriculum and will discover connections between classical and contemporary ideas in algebra, combinatorics, geometry, and topology.

A: I'm surprised nobody mentioned it, since it was written for this very purpose;
Problem Solving Strategies, by Arthur Engel
A great book in my opinion 
A: If you know a fair amount of calculus (and linear algebra) this might approachable: 
Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering by Steven H. Strogatz
And maybe these: 
Thirty-three Miniatures: Mathematical and Algorithmic Applications of Linear Algebra by Jiri Matousek
A Mathematical Mosaic: Patterns & Problem Solving  by Ravi Vakil 
A: Here are four suggestions:

*

*Arnold's problems book for school students, which is freely available via http://imaginary.org/sites/default/files/taskbook_arnold_en_0.pdf . Note that it is also available in many languages on the same web site.

*Martin Aigner, Günter Ziegler: Proofs from THE BOOK.

*A bit more advanced, but very good for students to look ahead what they might like to study later is Björn Engquist, Wilfried Schmid: Mathematics Unlimited.

*Samuli Siltanen: Step into the World of Mathematics.

A: Two excellent old books (but still very interesting) by great mathematicians. They are translated to several languajes:
Courant-Robbins, What is mathematics?
Rademacher-Toeplitz: Von Zahlen und Figuren (translated as "The Enjoyment of Math").
A: The list I give undergraduates and strong high schoolers is here. 
A: MacLane's Categories for the working mathematician 
Lang's Algebra.
Hartshorne's Algebraic geometry .
